Monday, October 21, 2019

Advanced Math Solutions - Series Convergence Calculator, Alternating Series Test

Last blog post, we discussed how to determine if an infinite p-series converges using the p-series test. In this blog post, we will discuss how to determine if an infinite alternating series converges using the alternating series test.

An alternating series is a series in the form ∑_{n=0}^∞(-1)^n∙a_n or ∑_{n=0}^∞(-1)^{n-1}∙a_n, where a_n>0 for all n. As you can see, the alternating series got its name from its terms that alternate between positive and negative values.

In some problems, you may come across alternating sign expressions that aren’t listed above. These various sign expressions can be simplified to equal the ones listed above. For example:

                  (-1)^{n+1}=(-1)^{n-1}∙(-1)^2=(-1)^{n-1}∙1=(-1)^{n-1}

Now that we know what an alternating series is, let’s discuss how to determine if the series converges, using the alternating series test.

Alternating Series Test:

An alternating series converges if all of the following conditions are met:
1. a_n>0 for all n
  • a_n is positive
2. a_n>a_(n+1)  for all n≥N,where N is some integer
  • a_n is always decreasing
3. lim_{n→∞} a_n=0

If an alternating series fails to meet one of the conditions, it doesn’t mean the series diverges. There are other tests that can be used to determine divergence.

Let’s see some examples of how to use the alternating series test to help you better understand.

First example (click here):

                                            ∑_{n=1}^∞\frac{(-1)^{n+1}}{\sqrt{n+1}}
1. Is a_n>0 for all n

                  ∑_{n=1}^∞\frac{(-1)^{n+1}}{\sqrt{n+1}}=∑_{n=1}^∞(-1)^{n+1}∙\frac{1}{\sqrt{n+1}}

                                                 a_n=\frac{1}{\sqrt{n+1}}>0 for all n

2. Is a_n>a_(n+1)  for all n≥N,where N is some integer?

                                   ∑_{n=1}^∞\frac{1}{\sqrt{n+1}}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+ ...
                                         \frac{1}{\sqrt{2}}>\frac{1}{\sqrt{3}}  and  \frac{1}{\sqrt{3}}>\frac{1}{\sqrt{4}}  and so on

                                                  a_n>a_(n+1)  for all n≥1

3. Is lim_{n→∞} a_n=0?

      Let’s use this property to determine the limit of a_n:


                                   lim_{n→∞}  \frac{1}{\sqrt{n+1}}=\frac{lim_{n→∞} 1}{lim_{n→∞} \sqrt{n+1}}=\frac{1}{∞}=0

4. Are all three conditions met?

                                                   Yes, the series converges.

Next example (click here):

                                                ∑_{n=1}^∞\frac{(-1)^n (n)}{n^2+2}

1. Is a_n>0 for all n?

                                                     a_n=\frac{n}{n^2+2}>0 for all n

2. Is a_n>a_{n+1}  for all n≥N,where N is some integer?

                   ∑_{n=1}^∞\frac{n}{n^2+2}=\frac{1}{3}+\frac{2}{6}+\frac{3}{11}+\frac{4}{18}  ...=\frac{1}{3}+\frac{1}{3}+\frac{3}{11}+\frac{2}{9}  ...

                                        \frac{1}{3}=\frac{1}{3},   \frac{1}{3}>\frac{3}{11},   \frac{3}{11}>\frac{2}{9}, and so on

                                                     a_n>a_(n+1)  for all n≥2

3. Is lim_{n→∞} a_n=0?

                                              lim_{n→∞}  \frac{n}{n^2+1}=lim_{n→∞}  \frac{\frac{1}{n}}{1+\frac{1}{n^2}}=\frac{lim_{n→∞}  \frac{1}{n}}{lim_{n→∞} 1+\frac{1}{n^2}}=  \frac{0}{1+0}=0

4. Are all three conditions met?

                                                     Yes, the series converges.

Last example (click here):

                                                 ∑_{n=2}^∞\frac{(-1)^{n+2}}{ln⁡(n)}

As discussed in the beginning, we can see a different alternating sign expression that wasn’t listed. However, we can simplify this expression to equal one that was listed.

                                    (-1)^{n+2}=(-1)^n*(-1)^2=(-1)^n

1. Is a_n>0 for all n?

                                                    a_n=\frac{1}{ln⁡(n)}>0 for all n

2. Is a_n>a_{n+1}  for all n≥N,where N is some integer?

                              ∑_{n=2}^∞\frac{1}{ln⁡(n)}=\frac{1}{ln⁡(2)}+\frac{1}{ln⁡(3)}+\frac{1}{ln⁡(4)}  ...

                                 \frac{1}{ln⁡(2)}>\frac{1}{ln⁡(3)},   \frac{1}{ln⁡(3)}>\frac{1}{ln⁡(4)},    and so on

                                                   a_n>a_{n+1}  for all n≥2

3. Is lim_{n→∞} a_n=0?

                                 lim_{n→∞}  \frac{1}{ln⁡(n)}=\frac{lim_{n→∞} 1}{lim_{n→∞} ln⁡(n)}=\frac{1}{∞}=0

4. Are all three conditions met?

                                                     Yes, the series converges.

The alternating series test has a lot of parts to it, but as long as you remember the three conditions, you’ll be able to master this topic! For more help or practice on the alternating series test, check out Symbolab’s Practice. Next blog post, I’ll go over the convergence test for telescoping series.

Until next time,

Leah

77 comments:

  1. Thanks for the shared this post with us. Shop today at Bernard Weatherill, an online luxury fashion store for men that offers a wide variety of formal, semi-formal, and casual options for men. Browse through countless categories and find all your favorite fashion brands at our online store.

    ReplyDelete
    Replies
    1. Hello everyone..Welcome to my free masterclass strategy where i teach experience and inexperience traders the secret behind a successful trade.And how to be profitable in trading I will also teach you how to make a profit of $12,000 USD weekly and how to get back all your lost funds feel free to email me on( [email protected] ) or whataspp number is +22999290178  
















      Hello everyone..Welcome to my free masterclass strategy where i teach experience and inexperience traders the secret behind a successful trade.And how to be profitable in trading I will also teach you how to make a profit of $12,000 USD weekly and how to get back all your lost funds feel free to email me on( [email protected] ) or whataspp number is +22999290178  
























      Delete
    2. Beautiful blog you got here.

      CYBER HACKING INTEL (CHI)
      Email-: [email protected]

      Have you ever wondered if it's possible to hack things the way hackers do it in the movies❓
      Yeah its possible!!!

      Hacking has been made simple in the word today, there are several hacking tools(Hacking Softwares and Hacking devices) out there that can be used to complete simple Hacking task.

      CYBER HACKING INTEL(CHI) is a group of skilled hackers that understand the use of this Hacking tools and can give you the best and secure Hacking services. Our services are fast and unique and guarantee a 💯 % chance of success.

      ❌Beware of Spammers and fake Hackers‼️ They are everywhere in the internet posting false ads and opening false sites. Do not be a victim of this guys, we am have come across so many of their victims.

      A couple of Hacking Services we offer are listed below-:
      🟢Phone Hacking, Cloning & Tracking.
      🟢To Sniff, Erase or Change a file in a Company, Organization e.t.c
      🟢BitCoin Mining.
      🟢Email & any Social Media account Hacking.
      🟢Generating Money by Sniffing Transactions.
      🟢Changing Records Such As-: Credit Score, School Grades, Criminal Records e.t.c.
      🟢Planting of Bugs/Malware/Spyware to monitor a specific target. And lots more.

      🔸if you need something other than what is listed above, you can give us an email ask if we offer the kinda Hacking Service you need.
      All these are what we can do for you. We are descrete and won't expose our service to you to anyone.

      Email-: [email protected]

      2019 © Cyber Hacking Intel.





      Delete
  2. Great blog! Thank you so much for sharing lovely post. I appreciate to blogger for helpful sharing.http://bit.ly/344ZdTT

    ReplyDelete
  3. Thanks for the best posting. I really appreciate it. Thanks a lot and keep sharing.
    <a href=" http://bit.ly/2pRJgSk> Movie Joker Celebrities Jackets</a>

    ReplyDelete
  4. Thanks for the best posting. I really appreciate it. Thanks a lot and keep sharing.
    Movie Joker Celebrities Jackets

    ReplyDelete
  5. I am looking for My assignment help Review admission to write admission essay. But with an increased in the demand for Services, many fake service providers have mushroomed all over the internet. There are many writing services which promise to be reliable and affordable. One such authentic service provider is Myassignmenthelp Review, So read review and take help from them and I got the best of me. But after reading edubirdie review I thought of writing this in order to make the students aware against this writing service so that they do not get trapped. The information that I have received from the reviews shed light on the poor quality of assignments. And In another ukessays review received from the student I found out that the company charges less but they compromise the work for the less amount of money. She had placed an order for doing her assignment on film review.

    ReplyDelete
  6. با توجه به اینکه شرکت نکست هر ساله آلبوم های جدید آلبوم کاغذ دیواری را به مجموعه خود اضافه می کند کاغذ دیواری هایی که مربوط به سال های گذشته کاغذ دیواری ایتالیایی هستند از جوری در می آیند و همین موضوع باعث می شود که روی این کاغذ دیواری ها

    تنها هتل جزیره هرمز با قیمت مناسب و امکانات خوب. هر سوییت شامل آشپزخانه و هرمز سرویس بهداشتی جداگانه. رخت خواب ها و حوله ها خیلی تمیز . غذای رستوران دریایی به سبک محلی و خیلی خوشمزه با قیمت مناسب.

    ReplyDelete
  7. Awesome blog. Actually am always confusing with series problems but after reading this article I cleared all my doubt's. Thanks!

    >>> http://ccexcelproperties.com/is-kodi-safe-to-use/

    ReplyDelete
  8. TruckingCube company is the best transportation company in India. Go to their website and see how it works.

    Moving Truck
    Truck Rental Near Me
    Bulk Transport

    ReplyDelete
  9. Very good solution. Geek Squad Tech Support has many new features to do their repair work effectively and efficiently. They are available 24/7 for all customers across the country. If you want any type of help, then definitely visit us.

    ReplyDelete
  10. Quotations and ends of sentences. It is traditional correct style in English grammar that all terminal punctuation comes inside quotation marks. My understanding is that this convention arose from the technical aspects of printing presses. Its purpose was to lessen the danger of breakage of fixed metal type in printing uk dissertation help online presses. In the era of electronic publishing of mathematics, this justification is obsolete, and we can replace the convention with logical punctuation.

    ReplyDelete
  11. With more than 10 years of experience in the industry of Custom Design Logo San Francisco and digital solutions, we confidently offer solutions of superior quality, with affordable packages to satisfy all customers. We also have a young, dynamic, and talented Creative Department willing to always work together with the client.

    ReplyDelete
  12. I have read your Excellent Post.This is Great Advanced Math Solutions.
    Online Essay Help UK

    ReplyDelete
  13. Your Blog is Good. I`m looking forward to more good blogs. In the meantime, I think you might like to visit my website too, which is about Legal Pads, Custom Printed Letter Pads, Letter Pads, Colored Legal Pads, Legal Notepad, Branded Notepads like stationery products.

    ReplyDelete
  14. really great blog about helping others from your blog.verywell put together. thanks.
    https://www.newamericanjackets.com/product/naruto-shippuden-uzumaki-naruto-jacket.html

    ReplyDelete
  15. The major portion of other up-front expenses is the deposit or binder you make apartment mortgage calculator at the time of the purchase offer and the remaining cash down payment you make at closing. In addition to the deposit and down payment.

    ReplyDelete
  16. Good to see your blog! i have noticed many amazing ideas and i tthink now you write a blog at Winter Fashion Trends Like Shearling Jackets or Leather Jackets For Winter!

    ReplyDelete
  17. Math is very important subject. It has great importance in universities. Teacher also focus on this subject because it is the most wanted subject. By: Assignment writing services.

    ReplyDelete
  18. This is additionally a generally excellent post which I truly appreciate perusing. It isn't regular that I have the likelihood to see something like this.
    live forex signals
    freeforex signals
    live signals
    live free forex signals
    gold forex signals
    daily forex signals
    free live forex signals
    gold forex signals
    live signals

    ReplyDelete
  19. Allassignmenthelp is your academic caretaker. Our online assignment assistance is one of the most sought after in the US and other countries and you should give us a chance to assist you. Write My Assignment

    ReplyDelete
  20. s128 apk merupakan aplikasi judi sabung ayam online yang bisa anda dapatkan dengan mendownloadnya, Pertarungan sabung ayam yang ada di situs ini tidak jauh berbeda dengan sabung ayam online s128 apk yang sudah banyak dikenal oleh pemain sabung ayam online.

    ReplyDelete
  21. Are you looking to hire top quality Custom Term Paper Help Services and cheap Legitimate Term Paper Writing Services? Rather than risking abject failure, you should consider hiring Legitimate Term Paper Help Provider.

    ReplyDelete
  22. win88 live casino adalah situs taruhan terbesar dan terpercaya di indonesia. Dimana kami adalah salah satu bandar judi bola online, casino online dan poker online terbaik yang mencakup seluruh bidang permainan game online

    ReplyDelete
  23. Pelayanan terbaik adalah prioritas utama dari Agen Judi idn poker Online. Kami menyediakan pelayanan customer service professional, sopan dan handal dalam bidangnya yang selalu siap membantu anda.

    ReplyDelete
  24. It isn't my first time to pay a brisk visit this site, I am visiting this site daily and get great information from here regular.
    Top SEO Companies in India
    Best SEO Companies in India
    Top SEO Company in India

    ReplyDelete
  25. Have you heard an explosion? That was my mind! Still, thanks for such boost of my mind activity. And now back to work, to read reviews of free logo makers

    ReplyDelete
  26. I’m really impressed with your article, thank you so much for sharing this amazing information. For a custom website design in delhi, visit OGEN Infosystem.
    Website Designing Company in Delhi

    ReplyDelete
  27. I appreciate your blog information about mathematics solutions. Basically I always follow online processes for math information and for that reason I have been using Asus router for internet connection.Recently I reset the router but after resetting asus router login not working and tried many processes but could not be login.If anyone has any solution then suggest to me.

    ReplyDelete
  28. Mathematics is a most wanted subject all over the World. Simple mathematics is easy, but advance mathematics is not so easy. So, for advanced mathematics solutions, students use calculators, but this is not a good step towards your expertness. If you want to become an expert in advanced mathematics, you have to leave using calculators and you have to use your mind. Dissertation writing service.

    ReplyDelete
  29. Replies
    1. The simple truth about binary options which many of us do not know is the fact that it is mainly based on predictions. Without proper knowledge of what next can happen to the stock market, you are sure to lose your funds. That is why it is important to be tutored or mentored by a professional trader in binary options. During few days of being mentored by Mrs Doris Ashley I've learnt much and also succeeding in trades and has been doing successful withdrawals and was able to recover all my lost funds. Feel free to contact her on [email protected] or whatsapp her +1 (516) 494 0313.for positive results

      Delete
  30. The simple truth about binary options which many of us do not know is the fact that it is mainly based on predictions. Without proper knowledge of what next can happen to the stock market, you are sure to lose your funds. That is why it is important to be tutored or mentored by a professional trader in binary options. During few days of being mentored by Mrs Doris Ashley I've learnt much and also succeeding in trades and has been doing successful withdrawals and was able to recover all my lost funds. Feel free to contact her on [email protected] or whatsapp her +1 (516) 494 0313.for positive results

    ReplyDelete
  31. watch your Ang Probinsyano Replay replays online in hd. All the replays are available for you in hd. Get all the updates and replays online.

    ReplyDelete
  32. There are many writing services that promise to be dependable and low-priced. the information that Cheapest Essays Now I've received from the opinions sheds mild on the negative first-class of assignments. My information is that this convention arose from the technical elements of printing presses.

    ReplyDelete
  33. I will carefully read the issues you just shared, I find there are some places that are not very clear
    mutilate a doll 2

    ReplyDelete
  34. This is very good content, i have been waiting for so many days for this type of content.thank u so much sir,you can visit my website for this type of content click here

    ReplyDelete
  35. There are several car hire companies in Bur Dubai who deal exclusively in upgraded and exotic cars and cater for the
    mobility need of both short-term visitors on vacations and business travelers. Some of these car hire companies in Bur
    Dubai location can be booked online right before you commence your flight to Bur Dubai. Otherwise, you may Car Rental from Bur Dubai location when you alight at the airport from representatives of these car hire companies.

    ReplyDelete
  36. When purchasing home health care directly from an individual provider (instead of an agency), it is even more important to conduct thorough screening. This should include an interview with the home health caregiver. You should also request references. Prepare for the interview by making a list of the older adult’s special needs. For example, the patient may require help getting into or out of a wheelchair. If so, the caregiver must be able to provide appropriate assistance.

    ReplyDelete
  37. When purchasing home health caredirectly from an individual provider (instead of an agency), it is even more important to conduct thorough screening. This should include an interview with the home health caregiver. You should also request references. Prepare for the interview by making a list of the older adult’s special needs. For example, the patient may require help getting into or out of a wheelchair. If so, the caregiver must be able to provide appropriate assistance.

    ReplyDelete
  38. I never really relied on a single service provider for all my academic requirements, unitil i found myassignmenthelp.com review. They have everything that makes an academic solution provider complete. I found myassignmenthelp when i was looking anywhere else for academic assistance. Notably there are service for all subjects.

    ReplyDelete
  39. We are best at what we do. MyAssignmentHelp.com has gained the reputation of being the best coursework help provider in Australia not because we successfully deliver premier quality help solution every time we receive a request from students but because we give something extra.

    ReplyDelete
  40. the best counterfeit money for sale

    Talking about
    choosing counterfeit money, one thing that comes to mind quality
    effectiveness and durability. the best counterfeit money for sale.
    But the best counterfeit money for sale isn’t pretended instead it’s
    like 100
    percent real currency. Most people United Nations agency area unit unaccustomed
    counterfeit cash usually escort a question wherever i able to purchase
    counterfeit
    cash. The answer is search on the net there area unit several
    corporations United
    Nations agency print counterfeit banknotes purchasable. the best
    counterfeit money for sale.
    Where to find the Best
    Counterfeit Money Online


    Many of these companies have a long experience in professionally selling this
    undetectable fake money of all major currencies of nations. Again, as per the
    need, one can Buy
    Dollars
    , Pounds, Euros, Yen, Yuan, Rayal, Rand and
    list is
    extensive. Interested person can buy counterfeit money online by placing the
    order and delivery of packages is done through trustworthy courier companies
    who safely deliver the orders within 3-5 working business days. the best
    counterfeit money for sale
    Packaging is completed with 100 percent expertness to stay the money
    safe and for convenience, big orders are converted into smaller ones.

    ReplyDelete
  41. Our customer care executives are available round the clock to guide you and in essay writing help. In case of an issue, all you need to do is contact us via call or chat options. Also, when you buying an essay from us, you purchase unlimited revisions free of cost.

    ReplyDelete
  42. Really great article, you guys are doing a great work, keep posting good stuff, If your audience is into Tech do share Best Instagram Story Saver Apps
    Gizdoc Tech Paradise

    ReplyDelete
  43. It was great reading your blog. Access 2 million course-specific study materials at our online learning library best assignment help for students

    ReplyDelete
  44. Take part in the online and offline auctions that are taking place every day.
    Banknilami in India

    ReplyDelete
  45. American Airlines Flights Reservations provide many facilities for the passengers where you can take advantage of those are facilities where you can find many facilities for your family members at the American Airlines Reservations official site.

    American airlines flights reservations

    ReplyDelete
  46. Bedava bonus almak icin en dogru adres neresi diye sorarsaniz orasi Bonusal sitesinden baska bir site olamaz

    ReplyDelete
  47. The whole study of crisis management can be split into core action plans and strategies. Being preventive is a modus operandi where preparedness and anticipation are key to reducing the likelihood of crisis occurring or at least minimizing its impact. Avail strategic crisis management help by our academic writers based in Australia.

    ReplyDelete